 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2018/1/d/solution

For any two distinct normalized Majorana fermion operators

η, χ

, let

f = (η + i χ) /2

. Then

{f, f^{†}} = \frac{1}{4} ({η, η} + {χ, χ}) = 1,

and

{f, f} = \frac{1}{4} ({η, η} - {χ, χ} + 2 i {η, χ}) = 0.

The same applies to

f^{†}

. Both specified pairs have this form. Moreover, the Majorana operators entering

f_{1}

are all distinct from those entering

f_{2}

, so every cross anticommutator is zero. Therefore

{f_{i}, f_{j}^{†}} = δ_{ij}, {f_{i}, f_{j}} = {f_{i}^{†}, f_{j}^{†}} = 0 .

Thus

f_{1}, f_{2}

are ordinary fermionic annihilation operators, constructed by a different pairing of the four Majorana operators.

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